In the strong driving range (\(\Omega _{p}>-13\mathrm {dBm}\) ), the peaks get broader and overlap with each other, eventually forming a lotus pattern, which is too complex to get the analytic solution. However, we can obtain the steady solutions of \(\rho _{11}\)  with \(\Delta =0\)  by
calculating the optical Bloch equations (see Appendix ),

11 =12-102n=-
1n212+(p/2-n)2, 

where \(\gamma _{1}=3\gamma _{10}/4+\gamma _{1}^{\phi }/2 \)  and n are integers. \(\Omega _{n}\)  is a series of Bessel functions with variable \(\Omega _{p}/\omega \)  [see Eq. (REF )]. When \(\tau \rightarrow 0\) , \(\Omega _{0}\rightarrow 1\)  and \(\Omega _{n\ne 0}\rightarrow 0\) , then Eq. () can be reduced to \(\rho _{11}\approx \frac{1}{2}-\frac{\gamma _{10}}{2}\frac{\gamma _{1}}{\gamma _{1}^{2}+(\Omega _{p}/2)^{2}}\) , which is close to the conventional result of the un-modulated system \(\rho _{11}=\frac{1}{2}-\frac{\gamma _{10}}{2}\frac{\gamma _{1}^{\prime }}{\gamma _{1}^{\prime }\gamma _{10}+\Omega _{p}^{2}}\)  with \(\gamma _{1}^{\prime }=\gamma _{10}/2+\gamma _{1}^{\phi }\) , except the effective Rabi frequency of the probe field to be \(\Omega _{p}/2\) . These indicate that when \(\tau \rightarrow 0\) , the central peak of the multi-peak phenomenon is similar to the single resonant peak in the un-modulated system. However, when \(\tau \)  away from 0, \(\Omega _{n}\)  significantly modifies the signal and induces a non-trivial phenomenon, coherent destruction of tunneling [1]}, [2]}, [3]}, where the steady excitation probability \(\rho _{11}\)  is partially suppressed and arising from the
superposition of degenerate Floquet states. When \(\Delta =0\) , from Eq. () one finds that the conditions for \(\rho _{11}\)  to take the local minimum values are \(\Omega _p=2n\omega ,~(n=1,2\cdots )\) . Expanding to the more general cases, the positions of the coherent destruction of tunneling are determined by the relationship between the effective Rabi frequency and the modulation frequency, i.e., \(\sqrt{\Omega _{p}^{2}+\Delta ^{2}}=2n\omega ,~(n=1,2\cdots )\) , as the white dashed lines shown in Fig. REF (b).
